Advanced Algorithms (IV) Chihao Zhang Shanghai Jiao Tong University - - PowerPoint PPT Presentation

Advanced Algorithms (IV)

Chihao Zhang

Shanghai Jiao Tong University

• Mar. 18, 2019

Advanced Algorithms (IV) 1/11

Review

MaxCut Input: An undirected graph G V E . Problem: A set S V that maximizes E S S .

Integer Program

max

e u v E

xuxv s.t. xu u V

Vector Program

max

e u v E

wT

uwv

s.t. wu

n

u V wT

uwu

u V

Advanced Algorithms (IV) 2/11

Review

MaxCut Input: An undirected graph G = (V, E). Problem: A set S ⊆ V that maximizes

• E(S, ¯

S)

• .

Integer Program

max

e u v E

xuxv s.t. xu u V

Vector Program

max

e u v E

wT

uwv

s.t. wu

n

u V wT

uwu

u V

Advanced Algorithms (IV) 2/11

Review

MaxCut Input: An undirected graph G = (V, E). Problem: A set S ⊆ V that maximizes

• E(S, ¯

S)

• .

Integer Program

max 1 2 ∑

e={u,v}∈E

(1 − xuxv) s.t. xu ∈ {−1, 1} , ∀u ∈ V

Vector Program

max

e u v E

wT

uwv

s.t. wu

n

u V wT

uwu

u V

Advanced Algorithms (IV) 2/11

Review

MaxCut Input: An undirected graph G = (V, E). Problem: A set S ⊆ V that maximizes

• E(S, ¯

S)

• .

Integer Program

max 1 2 ∑

e={u,v}∈E

(1 − xuxv) s.t. xu ∈ {−1, 1} , ∀u ∈ V

Vector Program

max 1 2 ∑

e={u,v}∈E

( 1 − wT

uwv

) s.t. wu ∈ Rn, ∀u ∈ V wT

uwu = 1,

∀u ∈ V

Advanced Algorithms (IV) 2/11

max 1 2 ∑

e={u,v}∈E

( 1 − wT

uwv

) s.t. wu ∈ Rn, ∀u ∈ V wT

uwu = 1,

∀u ∈ V is equivalent to a positive semi-definite programming (solvable in polynomial-time) Let wv v V be an optimal solution. Task: Round wv v V to a cut

Advanced Algorithms (IV) 3/11

max 1 2 ∑

e={u,v}∈E

( 1 − wT

uwv

) s.t. wu ∈ Rn, ∀u ∈ V wT

uwu = 1,

∀u ∈ V is equivalent to a positive semi-definite programming (solvable in polynomial-time) Let { wv}v∈V be an optimal solution. Task: Round wv v V to a cut

Advanced Algorithms (IV) 3/11

max 1 2 ∑

e={u,v}∈E

( 1 − wT

uwv

) s.t. wu ∈ Rn, ∀u ∈ V wT

uwu = 1,

∀u ∈ V is equivalent to a positive semi-definite programming (solvable in polynomial-time) Let { wv}v∈V be an optimal solution. Task: Round { wv}v∈V to a cut

Advanced Algorithms (IV) 3/11

Goemans–Williamson Rounding

• 1. Pick a random hyperplane crossing the origin;
• 2. The plane separates V into two sets.

Implementation

• 1. Choose a vector r

r rn where each ri i.i.d.

• 2. Let S

u V rTwu .

Advanced Algorithms (IV) 4/11

Goemans–Williamson Rounding

• 1. Pick a random hyperplane crossing the origin;
• 2. The plane separates V into two sets.

Implementation

• 1. Choose a vector r

r rn where each ri i.i.d.

• 2. Let S

u V rTwu .

Advanced Algorithms (IV) 4/11

Goemans–Williamson Rounding

• 1. Pick a random hyperplane crossing the origin;
• 2. The plane separates V into two sets.

Implementation

• 1. Choose a vector r = (r1, . . . , rn) where each ri ∼ N(0, 1) i.i.d.
• 2. Let S ≜

{ u ∈ V : rT wu ≥ 0 } .

Advanced Algorithms (IV) 4/11

Analysis

Proposition

r r is a point on Sn

uniformly at random.

Proposition

An edge u v E is separated with probability arccos wu

Twv .

Proposition

Random hyperplane rounding is a

• approximation of MaxCut.

Advanced Algorithms (IV) 5/11

Analysis

Proposition

r ∥r∥ is a point on Sn−1 uniformly at random.

Proposition

An edge u v E is separated with probability arccos wu

Twv .

Proposition

Random hyperplane rounding is a

• approximation of MaxCut.

Advanced Algorithms (IV) 5/11

Analysis

Proposition

r ∥r∥ is a point on Sn−1 uniformly at random.

Proposition

An edge {u, v} ∈ E is separated with probability 1

π arccos(

wu

T

wv).

Proposition

Random hyperplane rounding is a

• approximation of MaxCut.

Advanced Algorithms (IV) 5/11

Analysis

Proposition

r ∥r∥ is a point on Sn−1 uniformly at random.

Proposition

An edge {u, v} ∈ E is separated with probability 1

π arccos(

wu

T

wv).

Proposition

Random hyperplane rounding is a 0.878-approximation of MaxCut.

Advanced Algorithms (IV) 5/11

Qvadratic Program

We try to apply Goemans–Williamson rounding to general quadratic programs.

Qvadratic Program

max

i j n

ai jxixj s.t. xi i n We assume A ai j

i j n is positive semi-definite.

Advanced Algorithms (IV) 6/11

Qvadratic Program

We try to apply Goemans–Williamson rounding to general quadratic programs.

Qvadratic Program

max

i j n

ai jxixj s.t. xi i n We assume A ai j

i j n is positive semi-definite.

Advanced Algorithms (IV) 6/11

Qvadratic Program

We try to apply Goemans–Williamson rounding to general quadratic programs.

Qvadratic Program

max ∑

1≤i,j≤n

ai,jxixj s.t. xi ∈ {−1, +1} , i = 1, . . . , n. We assume A ai j

i j n is positive semi-definite.

Advanced Algorithms (IV) 6/11

Qvadratic Program

We try to apply Goemans–Williamson rounding to general quadratic programs.

Qvadratic Program

max ∑

1≤i,j≤n

ai,jxixj s.t. xi ∈ {−1, +1} , i = 1, . . . , n. We assume A = (ai,j )

1≤i,j≤n is positive semi-definite.

Advanced Algorithms (IV) 6/11

Rounding

We simply follow G-W…

Vector Program

max

i j n

ai jvT

i vj

s.t. vi

n

i n

• 1. Compute vi

i n.

• 2. Pick a vector r u.a.r on Sn

.

• 3. xi

if vi

Tr

; xi

• therwise.

Advanced Algorithms (IV) 7/11

Rounding

We simply follow G-W…

Vector Program

max ∑

1≤i,j≤n

ai,jvT

i vj

s.t. vi ∈ Rn, i = 1, . . . , n.

• 1. Compute vi

i n.

• 2. Pick a vector r u.a.r on Sn

.

• 3. xi

if vi

Tr

; xi

• therwise.

Advanced Algorithms (IV) 7/11

Rounding

We simply follow G-W…

Vector Program

max ∑

1≤i,j≤n

ai,jvT

i vj

s.t. vi ∈ Rn, i = 1, . . . , n.

• 1. Compute {

vi}1≤i≤n.

• 2. Pick a vector r u.a.r on Sn−1.
• 3. ˆ

xi = 1 if vi

Tr ≥ 0; ˆ

xi = −1 otherwise.

Advanced Algorithms (IV) 7/11

Analysis

Proposition

E xixj arcsin vT

i

vj

Proposition

Random hypergraph rounding is a

• approximation of QP.

Proof.

Use Schur producet theorem.

Advanced Algorithms (IV) 8/11

Analysis

Proposition

E [ ˆ xiˆ xj ] = 2 π arcsin(ˆ vT

i · ˆ

vj).

Proposition

Random hypergraph rounding is a

• approximation of QP.

Proof.

Use Schur producet theorem.

Advanced Algorithms (IV) 8/11

Analysis

Proposition

E [ ˆ xiˆ xj ] = 2 π arcsin(ˆ vT

i · ˆ

vj).

Proposition

Random hypergraph rounding is a 2

π -approximation of QP.

Proof.

Use Schur producet theorem.

Advanced Algorithms (IV) 8/11

Analysis

Proposition

E [ ˆ xiˆ xj ] = 2 π arcsin(ˆ vT

i · ˆ

vj).

Proposition

Random hypergraph rounding is a 2

π -approximation of QP.

Proof.

Use Schur producet theorem. □

Advanced Algorithms (IV) 8/11

Correlation Clustering

Given a undirected graph G V E in which each e E has two weights we we . Find a partition S Sk of V. E edge in a cluster; E edges between clusters. The goal is to maximize

e E

we

e E

we

Advanced Algorithms (IV) 9/11

Correlation Clustering

▶ Given a undirected graph G = (V, E) in which each e ∈ E has two weights w+

e , w− e ≥ 0.

Find a partition S Sk of V. E edge in a cluster; E edges between clusters. The goal is to maximize

e E

we

e E

we

Advanced Algorithms (IV) 9/11

Correlation Clustering

▶ Given a undirected graph G = (V, E) in which each e ∈ E has two weights w+

e , w− e ≥ 0.

▶ Find a partition S = (S1, . . . , Sk) of V. E edge in a cluster; E edges between clusters. The goal is to maximize

e E

we

e E

we

Advanced Algorithms (IV) 9/11

Correlation Clustering

▶ Given a undirected graph G = (V, E) in which each e ∈ E has two weights w+

e , w− e ≥ 0.

▶ Find a partition S = (S1, . . . , Sk) of V. ▶ E+(S) ≜ edge in a cluster; E−(S) ≜ edges between clusters. The goal is to maximize

e E

we

e E

we

Advanced Algorithms (IV) 9/11

Correlation Clustering

▶ Given a undirected graph G = (V, E) in which each e ∈ E has two weights w+

e , w− e ≥ 0.

▶ Find a partition S = (S1, . . . , Sk) of V. ▶ E+(S) ≜ edge in a cluster; E−(S) ≜ edges between clusters. ▶ The goal is to maximize ∑

e∈E+(S)

w+

e +

∑

e∈E−(S)

w−

e .

Advanced Algorithms (IV) 9/11

Vector Program

For k n, let ek be the k-th unit vector. max

u v E

wu v xT

uxv

wu v xT

uxv

s.t. xu e en u V

Relaxation

max

u v E

wu v xT

uxv

wu v xT

uxv

s.t. xT

vxv

v V xT

uxv

u v V xu

n

u V

Advanced Algorithms (IV) 10/11

Vector Program

For 1 ≤ k ≤ n, let ek be the k-th unit vector. max

u v E

wu v xT

uxv

wu v xT

uxv

s.t. xu e en u V

Relaxation

max

u v E

wu v xT

uxv

wu v xT

uxv

s.t. xT

vxv

v V xT

uxv

u v V xu

n

u V

Advanced Algorithms (IV) 10/11

Vector Program

For 1 ≤ k ≤ n, let ek be the k-th unit vector. max ∑

{u,v}∈E

( w+

u,v(xT uxv) + w− u,v(1 − xT uxv)

) s.t. xu ∈ {e1, . . . , en} , ∀u ∈ V.

Relaxation

max

u v E

wu v xT

uxv

wu v xT

uxv

s.t. xT

vxv

v V xT

uxv

u v V xu

n

u V

Advanced Algorithms (IV) 10/11

Vector Program

For 1 ≤ k ≤ n, let ek be the k-th unit vector. max ∑

{u,v}∈E

( w+

u,v(xT uxv) + w− u,v(1 − xT uxv)

) s.t. xu ∈ {e1, . . . , en} , ∀u ∈ V.

Relaxation

max ∑

{u,v}∈E

( w+

u,v(xT uxv) + w− u,v(1 − xT uxv)

) s.t. xT

vxv = 1,

∀v ∈ V, xT

uxv ≥ 0,

∀u, v ∈ V, xu ∈ Rn, ∀u ∈ V.

Advanced Algorithms (IV) 10/11

Rounding

Follow G-W and choose two hyperplanes.. We always obtain at most four clusters.

Proposition

Two random hyperplane rounding is a

• approximation for

correlation clustering.

Advanced Algorithms (IV) 11/11

Rounding

Follow G-W and choose two hyperplanes.. We always obtain at most four clusters.

Proposition

Two random hyperplane rounding is a

• approximation for

correlation clustering.

Advanced Algorithms (IV) 11/11

Rounding

Follow G-W and choose two hyperplanes.. We always obtain at most four clusters.

Proposition

Two random hyperplane rounding is a

• approximation for

correlation clustering.

Advanced Algorithms (IV) 11/11

Rounding

Follow G-W and choose two hyperplanes.. We always obtain at most four clusters.

Proposition

Two random hyperplane rounding is a 3

4-approximation for

correlation clustering.

Advanced Algorithms (IV) 11/11